Byrnes-Isidori-based dynamic sliding-mode control for nonminimum phase hypersonic vehicles

Aerospace Science and Technology 95 (2019) 105478

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Byrnes-Isidori-based dynamic sliding-mode control for nonminimum phase hypersonic vehicles Yuxiao Wang, Tao Chao, Songyan Wang, Ming Yang ∗ Control and Simulation Center, Harbin Institute of Technology, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 28 February 2019 Received in revised form 22 August 2019 Accepted 9 October 2019 Available online 14 October 2019 Keywords: Hypersonic vehicle Nonminimum phase Dynamic sliding-mode Byrnes-Isidori normalized form Internal dynamics

a b s t r a c t This paper deals with the dynamic sliding-mode (DSM) control problem for nonminimum phase hypersonic vehicles (HSVs). When the elevator is the only control surface available for the altitude dynamics, the HSV model exhibits unstable zero dynamics, preventing the application of standard inversion-based control techniques. To solve this problem, a DSM control method based on Byrnes-Isidori (B-I) normalized form is proposed, achieving asymptotic tracking of velocity and Flight-Path-Angel (FPA) while stabilizing internal dynamics. First, for the pitch dynamics with nonminimum phase behavior, external dynamics and internal dynamics are determined by coordinate transformation to convert the longitudinal model to B-I normalized form, based on which a criterion of nonminimum phase property is given by the stability analysis of internal dynamics. Then, a DSM control method is proposed for the FPA subsystem of nonminimum phase, which transforms the output tracking problem into stabilization problem of an augmented system consisting of internal dynamics and dynamic compensator, making closed-loop pole adjustable, and thus improves the tracking performance. The principle of parameters determination is proposed, which is proved to achieve the stability of the system on the sliding surface. Besides, nonlinear disturbance observer is utilized to compensate the error caused by dynamic inversion control. The proposed method is compared with approximate backstepping control and is shown to have superior tracking accuracy as well as robustness from the simulation results. This paper may also provide a beneficial guidance for control design of other complex systems of nonminimum phase. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Hypersonic air-breathing vehicles are sensitive to physical and aerodynamic parameters due to their high Mach numbers [1–3]. The design of guidance and control systems for air-breathing hypersonic vehicles presents a set of challenges which are unique to this class of aircraft. Since the emphasis is on the coupling of propulsion system and vehicle fuselage, the research on airbreathing hypersonic vehicle focuses on the control of velocity and altitude in longitudinal plane. One of the most difficult challenges encountered in designing flight control systems for HSVs is the nonminimum phase problem due to elevator-to-lift coupling [4]. If the dynamic inversion control is straight forwardly applied to HSVs, it results in exact tracking but the unstable zero dynamics remains an unstable part in the closed-loop system. Subsequently, the unstable zero dynamics of the non-minimum phase system hinder the application of standard inversion methods. Hence, this

*

Corresponding author. E-mail address: [email protected] (M. Yang).

https://doi.org/10.1016/j.ast.2019.105478 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

paper is mainly concerned with the issue of non-minimum phase characteristic on the hypersonic flight vehicle to improve the system reliability. In some ways, control methods are proposed based on a minimum phase hypersonic model with canards. The existence of canards transforms the pitch dynamics into a minimum phase system, and the conventional control methods can be applied [5–7]. However, the canard has been proved to have a more serious aerodynamic heat problem, which is not considered in these methods. The control methods for hypersonic vehicles without canards are still worth studying. Therefore, the control problem of HSVs without canards received more attention in recent years, and the control methods aiming at the nonminimum phase of HSVs have been investigated. Some linear adaptive control methods are presented, such as multiple model control [8], L1 control [9], integrated bilinear matrix inequalities and iterative control [10], robust stabilization approach via guardian maps theory [11], stable inversion approach [12], etc. However, the linear controllers suffer from the shortcomings that the linearized models around a trimmed point are considered while the nonlinearity terms are ignored and modeled as a bounded perturbation.

2

Y. Wang et al. / Aerospace Science and Technology 95 (2019) 105478

Therefore, more scholars devote themselves to the study of nonlinear control methods for nonminimum phase HSVs. Some beneficial works on nonlinear control have been done over the past few years. Representative ones among them are made by Parker [1, 13], Serrani [4,14] and Zong [15–17]. Firstly, as elevator-to-lift coupling is the cause of the nonminimum phase behavior of HSVs, strategically ignoring the elevator in the dynamic of FPA to obtain a full relative degree model is an obvious solution. A controloriented model was proposed by Parker [1], which has been widely used [13,15,18,19]. Input-output linearization of the approximate model can be achieved, based on which many control methods have been investigated, such as adaptive control [20,21] ans faulttolerant control [22]. Generally, this method will introduce an integral controller to offset the error caused by model approximation, which will degrade the dynamic tracking performance of the system. In addition, this method only works when the coupling is weak enough, i.e., a “slightly” nonminimum phase system [23]. The control law designed by the approximate model will result in instability when applied to the model with stronger nonminimum phase behavior [1]. Secondly, another research hotspot on the nonlinear control of non-minimum phase systems is output redefinition [4,14,16]. Its main idea is to perform an output redefinition such that the zero dynamics with respect to the new output are acceptable. Then, the problem of nonminimum phase is translated into defining a modified desired trajectory for the new output to track such that the original output tracks the original desired trajectory asymptotically. Correspondingly, the output redefinition method has two difficulties. On the one hand, there is no universal way to find a minimum phase output. On the other hand, the output redefinition method will bring new input uncertainty to the system. In view of these difficulties, some mature theories can be applied, such as flatness-based approach [24], stable inversion [25,26], etc. The problems of output determination and modified desired trajectory planning can be solved, but there are still some shortcomings. First, no systematic methods are proposed to find the minimum phase output of a complex system. Second, the output redefinition method will bring new input uncertainty to the system. Among the existing works on output redefinition, the Output-Redefinition-based Dynamic Inversion (ORDI) control in [16] covers relatively comprehensive approaches and convincing conclusions. It constructs a synthetic output to stabilize the internal dynamics. The output is a linear combination of the system output, an internal variable and integral tracking error, whose effect is very similar to Proportional-Integral (PI) control. The integral term can stabilize the internal dynamics, but it may reduce the dynamic tracking performance of the controller. In addition, some works on the aircraft attitude control provide new inspiration. Model transformation and adaptive sliding-mode techniques are applied to the aircraft attitude control [27–29]. In [27], nonminimum phase behavior of an under-actuated reentry vehicle is analyzed. Further, the vehicle attitude control system of nonminimum phase is transformed into a B-I normalized form, and the quantitative criterion of the nonminimum phase characteristic is given. In [28], a second-order sliding mode control method is proposed to stabilize the unstable internal dynamics of the system. However, the stability of the system on the sliding surface has not been proved strictly. Besides, the method needs the knowledge of the expected zero dynamics. Fortunately, it is found that output tracking can be achieved without knowledge of expected zero dynamics by a parameters constraint of sliding mode controller. For longitudinal trajectory tracking control with complete trajectory unknown, this B-I-based DSM control method can achieve excellent tracking effect. In this paper, an improved B-I-based DSM control method is proposed for nonminimum phase hypersonic vehicles. First, the nonminimum phase behavior of longitudinal dynamics model of

hypersonic vehicles is discussed. Second, external dynamics and internal dynamics are determined by coordinate transformation to convert the longitudinal model to B-I normalized form, based on which a criterion of nonminimum phase of hypersonic vehicles is given by the stability analysis of zero dynamics. Controllers are designed for velocity subsystem and FPA subsystem, respectively. For translational dynamics of minimum phase, a dynamic inversion sliding-mode controller is proposed. For the FPA subsystem of nonminimum phase, an improved B-I-based DSM control scheme is developed. The FPA tracking problem is transformed into stabilization problem of an augmented system consisting of internal dynamics and dynamic compensator when the system moves on the sliding surface. Parameters determination method of DSM is proposed to achieve the stabilization of augmented system without knowledge of expected zero dynamics. It is worth noting that the proposed method transfers the system uncertainty to the augmentation system of internal dynamics and dynamic compensator, which are only required to be stable, not accurate tracking. Therefore, output tracking performance and robustness are improved. Besides, nonlinear disturbance observer is used to compensate the error caused by dynamic inversion control. Simulations are presented to verify the effectiveness of the proposed controller. First, the comparison of tracking sinusoidal signal with approximate backstepping control is presented to show the superior tracking accuracy. Then, Monte Carlo simulations under parametric skewing are presented to verify the robustness. The remainder of this paper is organized as follows. In Section 2, the HSV longitudinal model is presented. In Section 3, the zero dynamics analysis is presented and the nonminimum phase criterion of HSVs is proposed. Then, the B-I-based DSM controllers for HSVs will be developed in Section 4. Next, simulations and discussions are given in Section 5. Finally, the conclusions are summarized in Section 6. 2. Model description In this section, the longitudinal dynamics model is established for an air-breathing HSV. In order to show the advantages of the proposed method more directly, FPA is designated as the output. The rigid-body longitudinal dynamics are given as follows [30]:

⎧ T cos(θ − γ ) − D ⎪ ⎪ V˙ = − g sin γ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ γ˙ = L + T sin(θ − γ ) − g cos γ ⎪ ⎪ θ˙ = Q ⎪ ⎪ ⎪ ⎪ ⎪ M ⎪ ⎪ ⎩ Q˙ =

mV

V

(1)

I yy

where

⎧   T ≈ q¯ S C T ,φ (α )φ + C T (α ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ D ≈ q¯ S C α 2 α 2 + C α α + C δe2 δ 2 + C δe δ + C 0 D D D D e D e  α δe 0 ⎪ ⎪ ¯ L ≈ q S C α + C δ + C ⎪ L L L e ⎪ ⎪  α2 2 ⎩ α α + C 0 + C δe δ  M ≈ z T T + q¯ S c¯ C M α + CM M M e

(2)

In the nonlinear model (1), state variables V , γ , θ , and Q are the velocity, FPA (Flight Path Angle), pitch angle, and pitch rate, respectively. T , D, L, and M are the thrust, drag, lift, and pitch moment, respectively. Control variables Φ and δe are the fuel throttle rate and elevator, respectively. The admissible ranges for variables are given in Table 1.

Y. Wang et al. / Aerospace Science and Technology 95 (2019) 105478

x = [x1 , x2 , · · · , xn ]T

Table 1 Admissible ranges for state, input, and variables of interest.

u = [u 1 , u 2 , · · · , um ]T

Symbol

Variables

Min

Max

V

Q

Velocity FPA Pitch angle Pitch rate

7500 ft/s −3 deg −5 deg −10 deg/s

11500 ft/s 3 deg 5 deg 10 deg/s

Φ δe

Fuel throttle rate Elevator

0.05 −20 deg

1.5 20 deg

α

Angle of attack Dynamic pressure

−5 deg 182.5 psf

5 deg 2200 psf

γ θ

q¯

mV

g V

cos γ

(3)

θ˙ = Q Q˙ =

(4) M

(5)

I yy

The output is the FPA γ , and the input is the elevator δe . Then, the model has a relative degree ρ = 1 < 3. The system inevitably has second order internal dynamics. The stability of the zero dynamics is analyzed under the trim condition. Let γ˜ = γ − γc , and bring lift expression into the FPA dynamics. The FPA error dynamics are obtained as follows. δ q¯ S (C Lα α + C Le δe + C L0 ) + T sin(θ − γ ) g γ˙˜ = − cos γ − γ˙c

mV

V

(6)

To obtain the zero dynamics of (6), the control law is designed as

δe =

1



δ

C Le

−

T sin α − mg cos γ mV

+ γ˙c

mV q¯ S

 − C Lα α + C L0 (7)

Clearly, we get that the error of FPA γ˜ is always 0 under control law (7). The dynamics of the system under control low (7) are the zero dynamics. A simple simulation is used to observe the zero dynamics of the system. The simulation shows that zero dynamics are unstable. The external output of the system is quickly beyond the scope, under the influence of unstable zero dynamics. The FPA subsystem is a nonminimum phase system. In Section 3.2, the nonminimum phase characteristics of the FPA subsystem are quantitatively analyzed based on the B-I normalized form theory. 3.2. Nonminimum phase criterion of air-breathing HSV A new evaluation criterion is proposed for the nonminimum phase property of the vehicle in this section. Without any loss of generality, the affine nonlinear control system is described as

x˙ = f (x) + g (x)u y = h ( x) where

T



T



T

h(x) = h1 (x), h2 (x), · · · , hm (x)

(9)

Here, we assume the system has a vector relative degree

[ρ1 , ρ2 , · · · , ρm ]T . Since the relative degree ρ of affine system (8)

Φ(x) : x → (ζ , η)

For FPA subsystem:

−



f (x) = f 1 (x), f 2 (x), · · · , f n (x)

is less than the number of states n, the system inevitably exist internal dynamics. In order to study the internal dynamics of the system more conveniently, it is hoped that the nonminimum phase system can be transformed into B-I normalized form by a coordinate transformation:

3.1. Analysis of nonminimum phase characteristic

L + T sin(θ − γ )

y = [ y 1 , y 2 , · · · , ym ]T

g (x) = g 1 (x), g 2 (x), · · · , gm (x)

3. Nonminimum phase property for air-breathing hypersonic vehicle

γ˙ =

3

(10)

where ζ and η denote external dynamics and internal dynamics, respectively. In the B-I normalized form of the system, control inputs are only included in external dynamics, but not in internal dynamics. The m dynamic equations of the normal form that depend explicitly on the control variables can be put in the following form:

ζ˙ ρ = b(ζ , η) + A (ζ , η)u

(11)

where ζ ρ ∈ R . If the matrix A (ζ , η) is locally nonsingular, the rigorous input–output feedback linearization is obtained as follows: m



u = A −1 (ζ , η ) −b(ζ , η) + v



(12)

Substitution of (12) into the normal form (8) yields

ζ˙ ji ˙i

= ζ ji+1

ζρi = v i η˙ = Γ (ζ, η) y i = ζ1i , j = 1, 2, · · · , ρi − 1, i = 1, 2, · · · , m

(13)

where η is the internal dynamics and v i is the pseudo control of the ith control channel. Since the affine nonlinear control system (8) is transformed into the normal form (13), there exists a similar coordinate transformation such that the rigid-body longitudinal model (1) of HSVs could be converted into a normal form like (13). The FPA subsystem is a nonminimum phase system due to elevator-to-lift coupling, which has unstable second-order zero dynamics θ and Q . In order to remove the control input in the internal dynamics, new internal dynamics are selected as a linear combination of natural internal dynamics and external dynamics, which can make the control input cancelled out in the new internal dynamics. Based on this idea, the subsystem (3)–(5) is transformed to a B-I normalized form as follows:

⎧ ⎪ ζ˙ = γ˙˜ ⎪ ⎪ ⎪ ⎨ η˙ 1 = θ˙ ⎪ I yy mV ˙ ⎪ ⎪ Q˙ − γ˜ ⎪ ⎩ η˙ 2 = δe δ q¯ ScC M q¯ SC Le

(14)

where the external dynamics

(8)

ζ = γ˜ and the internal dynamics

(15)

4



Y. Wang et al. / Aerospace Science and Technology 95 (2019) 105478

η1 = η2



θ Q −

I yy δe

q¯ ScC M

 (16)

γ˜

mV δ q¯ SC Le

The selection scheme aims at eliminating the control inputs in the internal dynamics, which is not unique. A more intuitive choice is made in this paper to eliminate the control quantity. Next, a criterion theorem for nonminimum phase characteristic of the model is proposed. Theorem 1. If the hypersonic vehicle system parameters satisfy (17), the FPA subsystem of the hypersonic vehicle is a nonminimum phase system. δe

CM

 α C

M

δ

C Me



C Lα

−

>0

δ

C Le

(17)

Proof. Bring the model (3)–(5) into (14), the dynamics of internal dynamics can be written as (13), (14): δ

q¯ ScC Me

η˙ 1 =

I yy

δ

η2 +

mV cC Me

(18)

ζ

δ

I y y C Le

 α  α CM C Lα CM C Lα η˙ 2 = − δ − δ ζ + δ − δ η1 + Γ C Me

C Le

C Me

(19)

C Le

which bring the tightly coupled, highly nonlinear and notoriously uncertain nature of HSV dynamics. Internal perturbation, external disturbance and channel coupling need to be observed, which can be added to the controller [31,32]. Then the inhibition of the disturbance will be achieved. The rigid-body longitudinal model of HSVs with uncertainty is considered:

⎧ T cos(θ − γ ) − D ⎪ ⎪ V˙ = − g sin γ + d1 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ γ˙ = L + T sin(θ − γ ) − g cos γ + d2 mV

V

⎪ ⎪ θ˙ = Q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Q˙ = M + d3 ⎩

(23)

I yy

where d = [d1 , d2 , d3 ]T are the unknown disturbances. Assumption 1. The disturbance d is continuous and satisfies

 j  d d    dt j  < μ where

(24)

μ > 0, j = 0, 1, 2.

where

Γ =

zT δ

q¯ ScC Me

−

 α C

M

δe

CM

T sin α − mg cos γ

−

δ

−

α

CL

q¯ SC Le

γc +

δe

CL

mV δ

q¯ SC Le

−

γ˙c

C Lα δ

C Le

2

+

α CM

A second order Disturbance Observer (DO) is proposed for the model (23) as follows [31]:

0 α2 + C M

δ

C Me (20)

Consider the zero dynamics of the system, i.e. the external dynamic ζ is 0. Again derivation of (19), we obtain: δ

η¨ 2 =

q¯ ScC Me

 α C

M

δ

I yy

C Me

−

C Lα



δ

C Le

η2 + Γ˙

(21)

If the hypersonic vehicle system parameters satisfies (17), then δ

q¯ ScC Me I yy

 α C

M

δe

CM

−

C Lα δe

CL

>0

(22)

The system (21) is unstable. As time increase, the internal state η2 satisfy limt →∞ η2  = ∞. The internal dynamics are unstable, so the system (14) is a nonminimum phase system. The theorem provides a quantitative criterion for the analysis of the nonminimum phase hypersonic vehicle model about the aerodynamic parameters. 2 Remark 1. The elevator produces pitching moment while generating the lift, then the pitching moment will change the angle of attack of the vehicle. The lift produced by this part of angel of attack is opposite to the lift direction produced by the elevator, and the former is greater than the latter when inequality (17) holds. Therefore, direct control of FPA with elevator will make inputs and states saturated rapidly, which is the physical meaning of Theorem 1. 4. Controller design considering nonminimum phase and model uncertainty 4.1. Nonlinear disturbance observer Compared with the traditional vehicles, hypersonic vehicles (HSV) have a faster speed and more extreme flight conditions,

⎧ ⎪ dˆ = p 11 + L 11 x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p˙ = − L  f (x) + dˆ + d˙ˆ 11 11

(25)

⎪ ˆ ⎪ ⎪ d˙ = p 12 + L 12 x ⎪ ⎪ ⎪  ⎩ p˙ 12 = − L 12 f (x) + dˆ ˆ

where dˆ and d˙ are estimates of d and d˙ respectively, p 11 and p 12 are auxiliary variables, and L 11 , L 12 are user chosen constants. L 11 = diag(l1ii ), l1ii > 0, L 12 = diag(l2ii ), l2ii > T cos(θ−γ )− D 0. f (x) = [ f 1 (x), f 2 (x), f 3 (x)]T . f 1 (x) = − g sin γ , m f 2 ( x) =

L + T sin(θ−γ ) mV

−

g V

cos γ , f 3 (x) =

M I yy

.

Let the estimation errors be defined as



e˜ = d˜

˙

d˜

T

(26)

d˜ = d − dˆ

(27)

˙ ˆ d˜ = d˙ − d˙

(28)

From (25)

˙ ˆ dˆ = L 11 d˜ + d˙

(29)

Subtracting both sides of (29) from d˙

˙

ˆ

d˜ = − L 11 d˜ + d˙ − d˙

˙ = − L 11 d˜ + d˜

(30)

Working on similar lines:

¨

d˜ = − L 12 d˜ + d¨

(31)

The observer error dynamics can be expressed in compact form as

e˙˜ = D e˜ + E d¨ where

(32)

Y. Wang et al. / Aerospace Science and Technology 95 (2019) 105478



D=

− L 11 I − L 12 0



(33)

0 I

E=

4.3. Controller design for pitch dynamics of nonminimum phase For the model of FPA subsystem with uncertainty as follows:

(34)

From (32)–(34), it is easy to see that it is always possible to select L 11 and L 12 such that the eigenvalues of D can be placed arbitrarily. Assuming that L 11 and L 12 are chosen in such a way that the eigenvalues of D are in the LHP, it is always possible to find a positive definite matrix P such that

DT P + P D = − Q

L + T sin(θ − γ )

γ˙ =

T

mV

(38)

λm

Let

2 P E μ

(39)

λm



1

v−

δ

C Le

T sin α − mg cos γ mV

4.2. Controller design for translational dynamics of minimum phase

γ˙˜ = v + d˜ 2

(50)

where v is the virtual control, and d˜ 2 = d2 − dˆ 2 . According to (18) and (19), it can be concluded that the B-I normalized form of FPA subsystem is written as follows:

ζ˙ = v + d˜ 2



η˙ 1 A 11 = η˙ 2 A 21



A 12 A 22

where A 11 = 0, A 12 = δe

mV cC M

For the model of velocity subsystem with uncertainty as follows:

− g sin γ + d1

(40)

Conventional dynamic inversion control method is adopted. Let V˜ = V − V c , we obtain the error dynamics of velocity:

˙

V˜ =

q¯ S cos α [C T ,φ (α )φ + C T (α )] − D m

mV + γ˙c − dˆ 2 q¯ S (49)

where dmax is the upper bound of estimation error of DO.

m

− γ˙c + d2 (48)

We obtain

2 P E μ

T cos(θ − γ ) − D

mV

 − C Lα α + C L0

(37)

where λm the smallest eigenvalue of Q . Therefore, after a sufficiently long time, the norm of the estimation error is bounded by

V˙ =

δ

q¯ S (C Lα α + C Le δe + C L0 )

+

The dynamic inversion controller is designed as

T

dmax =

(47)

T sin α − mg cos γ

δe =

≤ −λm e˜ 2 + 2 P E e˜ μ  ≤ −e˜  λm e˜  − 2 P E μ

+ d3

I yy

One obtains:

≤ −e˜ Q e˜ + 2 P E e˜ μ

(45)

Let γ˜ = γ − γc , we obtain the error dynamics of FPA:

(36)

T T V˙ (e˜ ) = e˜ D T P + P D e˜ + 2e˜ P E d¨

cos γ + d2

V

(46) M

Q˙ =

γ˙˜ =

V (e˜ ) = e˜ P e˜

g

−

mV

θ˙ = Q

(35)

Defining a Lyapunov function

e˜  ≤

5

− g sin γ − V˙ c + d1 (41)

δ

, B 2 = −(

I y y C Le 2 C α α 2 +C 0 M

M

δ

C Me

−(

Cα

M

δ

C Me

α CM δ

C Me

−

− Cα L

δ

C Le

Notice that d = −



η1 0 + η2 Γ δ

q¯ ScC Me I yy C Lα δ

C Le

mV δ q¯ SC Le

+

, A 21 =

), Γ =

)γc +

mV δ q¯ SC Le

d2 +

α CM δ

C Me

zT δ q¯ ScC Me

−

B1 B2

−

(51)

(52)

ζ C Lα δ

C Le

, A 22 = 0, B 1 =

T sin α −mg cos γ δ

q¯ SC Le

−

C Lα δ

C Le

+

γ˙c + d . I yy δ

q¯ ScC Me

d3 + d4 . d2 and d3 are the

uncertainties of longitudinal translational and rotational channels, respectively. d4 is a non-linear additional term caused by V˙ . Assumption 2. For constant commands, V˙ c and γ˙c are regarded as zero. For varying commands, V˙ c and γ˙c are bounded.

Sliding surface of subsystem (41) can be defined as

S 1 = V˜

(42)

The dynamic inversion controller is designed as

φ=

m q¯ S cos α C T ,φ (α )

−

−λ1 · Sign( S 1 ) + g sin γ + V˙ c − dˆ 1

q¯ S cos α C T (α ) − D

m

(43)

where dˆ 1 is the estimate of d1 , λ1 is a positive constant. We obtain

S˙ 1 = −λ1 · Sign( S 1 ) + d˜ 1

(44)

where d˜ 1 = d1 − dˆ 1 . Obviously, the controller can achieve the asymptotical convergence of the system. A complete stability analysis will be given in Section 4.4.

Under Assumption 1 and Assumption 2, it can be known that the nonlinear term

Γ in (52)

is bounded. A 11 A 12 The matrix is non-Hurwitz because of the nonA 21 A 22 minimum nature of the system. If the output is directly controlled by the virtual control input v, the internal dynamics of the system will quickly diverge. The control input will be saturated quickly and the output of the system will not be stable. Hence, it is revealed that the dynamic inversion control cannot be designed for the nonminimum phase system. To stabilize the internal dynamics of the closed-loop nonminimum phase system, a state feedback control can be used to make the eigenvalues of state matrix of the subsystem (51) and (52) in the LHP. However, the drawback of this method is that the equilibrium point of the system varies with the expected output when the gain parameter is constant. If the expected output of the system is time-varying, the tracking effect of the controller will be poor or even divergent. In order to improve the tracking performance of time-varying

6

Y. Wang et al. / Aerospace Science and Technology 95 (2019) 105478

Consider the new system for internal dynamics (52) and dynamic compensator χ











η˙ 1 A 11 A 12 η1 0 = + η˙ 2 A 21 A 22 η2 Γ χ˙ = F χ + G 1 ζ + G 21 η1 + G 22 η2



+

B1 B2

(57)

ζ

(58)

Substituting (56) into (57) and (58), the system of internal dynamics and the dynamic compensator χ motion on the DSM S 2 = 0 is derived as

⎡ ⎤ ˙ η1 ⎢ ⎣ η˙ 2 ⎦ = ⎢ ⎣ χ˙

trajectory, a B-I-based DSM control considering internal dynamics design is accomplished as follows. A dynamic sliding-mode (DSM) S is defined as a linear operator and is given as

χ˙ = F χ + G 1 ζ + G 2 η S = C χ + H ζ + C˜ η

(53)

where χ denotes a dynamic compensator, ζ and η denote external dynamics and internal dynamics, respectively. Parameters F , G 1 , G 2 , C , H , and C˜ are to be determined. The DSM is defined as S = 0, which solves the problem of internal dynamic stabilization for a class of nonminimum phase systems with relative order of 1. In [33], the linear subspace decomposition technique is combined with dynamic sliding mode control to solve attitude control problem, in which the parameters design process is complex. Consider the B-I normalized form of the subsystem (51) and (52), where ζ is the output, η1 , η2 are the internal dynamics, and v is the virtual control input. A B-I-based DSM control approach with relatively simple parametric design principle is proposed to solve the longitudinal trajectory tracking problem of HSVs. A schematic diagram is as follows to show the whole idea of DSM the control method. The hypersonic vehicle system is transformed to a B-I normalized form, and FPA is directly controlled by the DSM controller based on internal and external states feedback (Fig. 1). The DSM of subsystem (51) and (52) can be represented as

χ˙ = F χ + G 1 ζ + G 21 η1 + G 22 η2 S 2 = C χ + H ζ + C˜ 1 η1 + C˜ 2 η2

(54)

Theorem 2. Construct matrix

⎡ ⎢

P =⎢ ⎣

˜

− B 1HC 1

B 2 C˜ 1 H

+ A 21

G 21 −

G 1 C˜ 1 H

˜

− B 1HC 2 + A 12 B 2 C˜ 2 H ˜ G 22 − G 1HC 2

− BH1 C B2C H

F−

⎤ ⎥ ⎥ ⎦

(55)

G1C H

H

(C χ + C˜ 1 η1 + C˜ 2 η2 )

B2C H

+ A 21

G 21 − 0

H

B 2 C˜ 2

⎤

G 1 C˜ 1 H

G 22 −

G 1 C˜ 2 H

F−

G1C H

⎤

⎡ ⎤ ⎥ η1 ⎥ ⎣ η2 ⎦ ⎦

χ

(59)

0 Asymptotic convergence is achieved, as state matrix of the new system is P , which is Hurwitz. According to the condition of C˜ 1 = C GF21 , C˜ 2 = C GF22 and H = C G1 , F

ζ=

(54) can be solved to:

χ˙ σ

(60)

where σ = G 1 − FCH . As the nonlinear term Γ is bounded, there are:

|χ˙ | < κ

(61)

where κ is a positive constant. The external dynamic of the system (51) and (52) will converge to

  κ  ζ <   σ

(62)

with the internal dynamics stable. Consider the ideal case when tracking constant commands and the disturbance d2 and d3 tending to constant value as time increase. Under Assumption 1 and Assumption 2, the extended states of the system x = point x∗



η1 η2 χ

P −1 Γ , where

T



will converge to the equilibrium

T

, and ζ = σ0 = 0. However, when γc is a time-varying value, σ is a precision adjustment parameter, which will affect both the dynamic tracking accuracy and the robustness of the system. As σ increases, the system tracking performance improves, while the Hurwizt condition of matrix P requires that the system state corridor become narrow and the robustness of the system decreases. The σ needs to be adjusted to compromise the tracking performance and robustness of the system in different missions. 2

=

Γ = 0 Γ

0

C˜ 1 −

C G 21 F





η1 + C˜ 2 −

C G 22 F



η2 = 0

If the parameters do not satisfy the condition (1) C˜ 1 =

Proof. If the system is on the DSM S 2 = 0, the external dynamic

1

H

− BH1 C

+⎣Γ ⎦



C G1 F

The external dynamic ζ will achieve asymptotic convergence as time increase, when the system moves on the DSM S 2 = 0.

ζ =−

B 2 C˜ 1

− B 1HC 2 + A 12

Remark 2. The convergence of the external dynamic to zero requires that χ˙ = 0, S 2 = 0 in (54), we obtain

where A 12 , A 21 , B 1 , B 2 are defined in (52). If the parameters in (54) satisfy (1) C˜ 1 = C GF21 , C˜ 2 = C GF22 and H = (2) The matrix P is Hurwitz.

− B 1HC 1

⎡

Fig. 1. Schematic diagram of DSM control method.

˜

˜

⎡

(56)

(63) C G 21 , F

C˜ 2 = and H = there will be a fixed functional relationship between the two equilibrium points of internal dynamics. Considering the inherent characteristics of the system, the internal dynamics will not maintain the same linear relationship at different velocities, FPAs, etc. Even if the corresponding parameters are set for each working condition, the internal dynamics will not have C G 22 F

C G1 , F

Y. Wang et al. / Aerospace Science and Technology 95 (2019) 105478

an invariable linear relationship due to the existence of uncertainties, and the system output ζ will not converge to 0. It is worth mentioning that this is an inherent feature of this form of DSM, which is independent of the system and the derivation method. Any DSM in this form which does not satisfy condition (1) cannot achieve precise output tracking.

Table 2 Initial condition. Variable

Value

Variable

Value

V h q

7860 ft/s 110000 ft 0 0

S c

17 ft2 17 ft 0.008 rad 0.008 rad

γ

Remark 3. In system (57), equilibrium point is x∗ = P −1 Γ . Because of the existence of uncertainty and the change of velocity, P and Γ are not constant matrices. Generally, the velocity will not change rapidly during cruise. If the parameters make all P in the range of variation satisfy the Hurwitz condition, variability of P and Γ have little effect on the performance of closed-loop system. For d2 and d3 in Γ , it can only make their coefficients in the non-linear term as small as possible by choosing appropriate internal dynamics, which cannot be fully compensated. This problem cannot be solved in the back-stepping or other methods either. As control variables exist in both inner and outer loops of the system, the compensation of the disturbance in the inner loop will always pollute the outer loop. In the case of complete trajectory unknown, it is the performance limitation for closed-loop systems of nonminimum phase, which is due to the nature of the system, not the controller. Differentiating S 2 in (54), we obtain

S˙ 2 = C χ˙ + H ζ˙ + C˜ 1 η˙ 1 + C˜ 2 η˙ 2

(64)

v =−

1 H

λ2 · Sign( S 2 ) + C χ˙ + C˜ 1 η˙ 1 + C˜ 2 η˙ 2



(65)

The main aim of this section is to design the control law v in (51) and parameters in (54) so that the nonminimum phase subsystem (51) and (52) is asymptotically stable. In Section 4.4, we will analyze the stability of the system (1).

5. Simulation results In this section, the results of numerical simulations are presented to demonstrate the effectiveness and applicability of the controller design method proposed in Section 3. The initial conditions of the hypersonic vehicle are given in Table 2. First, parameter in criterion (17) is derived as follows: δ

 α C

M

δ

C Me

−

C Lα δ

F = −49.82, G 21 = −42.70, C˜ 1 = −49.97,

λ1 = 20,

Sign( S ) =

V˙ L = S 1 · S˙ 1 + S 2 · S˙ 2

 = S 1 · −λ1 · Sign( S 1 ) + d˜ 1 + S 2 · (C χ˙ + H ζ˙ + C˜ 1 η˙ 1 + C˜ 2 η˙ 2 )  = S 1 · −λ1 · Sign( S 1 ) + d˜ 1   1 ˙ + S 2 · C χ + H − λ2 · Sign( S 2 ) + C χ˙ + C˜ 1 η˙ 1 + C˜ 2 η˙ 2 H + d˜ 2 + C˜ 1 η˙ 1 + C˜ 2 η˙ 2

  = S 1 · −λ1 · Sign( S 1 ) + d˜ 1 + S 2 · −λ2 · Sign( S ) + H · d˜ 2  < | S 1 | · (−λ1 + dmax ) + | S 2 | · −λ2 + | H | · dmax (67)

G 22 = 64.10, H = 42.24,

(69)

C˜ 2 = 75.02,

λ2 = 10

In order to prevent system chattering, the sign function Sign( S ) is improved to

(66)

Proof. Choosing the following Lyapunov function candidate V L = 1 2 S + 12 S 22 , differentiating V L with respect to time t yields 2 1

(68)

G 1 = 36.19,

C = −58.31,

the output of the system (1) will be asymptotically stable.

= 15.75 > 0

According to the Theorem 1, it can be seen that the FPA subsystem of the hypersonic vehicle is a nonminimum phase system. The proposed method is used to control the hypersonic vehicle system. The control parameters in (43), (54) and (65) are given as

In this section, for the system (1) stability proofs have been discussed.

λ1 > dmax λ2 > | H | · dmax



C Le

4.4. Stability analysis of closed-loop system

Theorem 3. Under the DSM (42), (54) and the control law (43), (65), if the parameters satisfy

α θ

As λ1 > dmax and λ2 > | H | · dmax , V˙ L < 0. The DSM (42) and (54) are reachable. Obviously, the subsystem (41) is stable as V˜ = S 1 = 0. According to Theorem 2, the subsystem (51) and (52) will be asymptotically stable in S 2 = 0. It can be concluded that the DSM-based control scheme designed for the closed-loop system (1) could stabilize the nonminimum phase system. 2

CM

From (51) and (64), we obtain the control law

7

⎧ ⎪ ⎨ ⎪ ⎩

S , S + 0 .1

S >0

0,

S =0

−S , S − 0 .1

S <0

(70)

Choosing the output commands

V c = 7080, ft/s

(71)

γc = 0.03 sin(30t /π ), rad

(72)

In order to test the control performance of the nonminimum phase subsystem, the FPA tracks a sinusoidal signal with an amplitude of 0.03 and a period of 60 s. The backstepping controller proposed in [15] is used as a comparison, which is designed based on control-oriend-model (COM). The control method strategically ignoring the elevator in the dynamic of FPA to obtain a full relative degree model, and the standard inversion-based control techniques are applied. The tracking curves are shown in Fig. 2. The controller 1 is the one proposed in this paper, and the controller 2 is back-stepping controller in [15]. It can be seen that the tracking performance of velocity loop is similar, and the tracking performance of the proposed method in FPA loop has obvious advantages.

8

Y. Wang et al. / Aerospace Science and Technology 95 (2019) 105478

Fig. 2. Responses of the velocity and FPA.

Fig. 3. Responses of the internal dynamics.

Fig. 4. Responses of the control input. Table 3 Uncertainty parameters model. Parameter class

Distribution

Range of error

Aerodynamic coefficient Aerodynamic moment coefficient Atmospheric density

Average distribution Average distribution Average distribution

20% 20% 20%

The internal dynamics and control input curves are shown in Fig. 3 and Fig. 4, respectively. The internal dynamics and control input are within a reasonable range without saturation or divergence. In order to verify the robustness of the controller, 1000 groups of Monte Carlo simulations are carried out. An uncertain parameter model for hypersonic vehicle Monte Carlo simulation is given in Table 3. Choosing the output commands

V c = 7080, ft/s

(73)

γc = 0.03 sin(25t /π ), rad

(74)

Simulation results are as follows (Fig. 5). The output error frequency histogram of the system is as follows (Fig. 6). Monte Carlo simulation results show that the system is robust. The final velocity tracking errors concentrate within 0.5 and the maximum value does not exceed 0.8. The final FPA tracking errors

converge to a sufficiently small value. The states are stable after the initial transient oscillation, and eventually stabilize at the equilibrium point of different aerodynamic parameters. The internal dynamics have been effectively stabilized. The increase of velocity tracking error at about 25 s is due to the unpowered acceleration of the HSV caused by the descending trajectory, which is due to the inherent characteristics of the system. The simulation results show that the designed controller can effectively stabilize the unstable internal dynamics of the system while ensuring the tracking effect of the external state. The comparison of tracking sinusoidal signal with approximate backstepping control show that the proposed controller achieve the superior tracking accuracy. Monte Carlo simulations show that the system can keep high precision tracking under parametric deviation of 20%. 6. Conclusions A DSM control method based on B-I normalized form is proposed. The FPA subsystem is converted to B-I normalized form, based on which a criterion of nonminimum phase characteristic is given by the stability analysis of internal dynamics. An improved B-I-based DSM controller is developed, achieving asymptotic tracking of outputs while stabilizing internal dynamics. The salient features of the proposed approach consist in presenting a quantitative criterion for the minimum phase characteristics of HSVs based on B-I normalized form and transforming

Y. Wang et al. / Aerospace Science and Technology 95 (2019) 105478

9

Fig. 5. Results of the Monte Carlo simulations. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

Fig. 6. Error frequency histogram.

the output tracking problem into stabilization problem of an augmented system consisting of internal dynamics and dynamic compensator, transferring uncertainty to a stabilization system with relatively loose requirements. The improved parameter determination method eliminates the drawback that the desired internal dynamics must be known. When the complete trajectory information is unknown, this control method can achieve relatively excellent tracking performance. It confirms the advantages of the proposed method by contrast simulations and Monte Carlo simulations. In future research, multiple sets of parameters will be designed as the state of the system changes. The application of gain scheduling will make the performance and robustness of the system to a higher level. Declaration of competing interest The authors declare no conflict of interest. Acknowledgements The research presented in this document is supported by the National Natural Science Foundation of China (Grant numbers 61403096, 61627810, 61790562). References [1] J.T. Parker, A. Serrani, S. Yurkovich, et al., Control-oriented modeling of an airbreathing hypersonic vehicle, J. Guid. Control Dyn. 30 (2007) 856–869. [2] W.M. Bao, Present situation and development tendency of aerospace control techniques, Acta Autom. Sin. 39 (2013) 697–702. [3] C.I. Byrnes, A. Isidor, J.C. Willems, Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems, IEEE Trans. Autom. Control 36 (2002) 1228–1240.

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